# What does the bar above the integral mean?

I have seen this in textbooks, and indeed found this Stack thread about how to create the bar in LaTEX, but don't know what it means:

$\overline\int,\underline\int$

• In textbooks, they probably introduce/explain the notation...? It's likely to be an upper (Darboux) integral (and with a bar below for the lower integral); see here. May 10 '17 at 14:23
• Could they be Darboux integrals? May 10 '17 at 14:24
• Ah thank you, turns out it was Riemann integrals which is the same thing :) May 10 '17 at 14:27
• As a side note, sometimes $\overline{\lim} a_n$ and $\underline{\lim} a_n$ are used to denote limit superior and limit inferior respectively. May 10 '17 at 16:27

These are denoted as the upper and lower Riemann integrals respectively.

More so, we can construct the following:

Let $P=\left\{x_0,x_1,...,x_n\right\}$ be a partition on $[a,b]$ for $n\in \mathbb{N}$.

Notating the lower and upper sums of some function $f$ with respect to its partition $P$, as $L(P,f)$ and $U(P,f)$, we can define the following:

$$\underline {\int_a^b}f=sup\left\{L(P,f):\forall \ partitions \ P \ on \ [a,b]\right\}$$

$$\overline {\int_a^b}f=inf\left\{U(P,f):\forall \ partitions \ P \ on \ [a,b]\right\}$$

Furthermore, we note that for a reimann integral to exist on some bounded interval $[a,b]$,

$$\underline {\int_a^b}f=\int_a^bf=\overline {\int_a^b}f$$